Extremal graphs for homomorphisms

Extremal graphs for homomorphisms

The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of ho...

Extremal Graphs for Homomorphisms II

Extremal problems for graph homomorphisms have recently become a topic of much research. Let hom(G,H) denote the number of homomorphisms from G to H. A natural set of problems arises when we fix an image graph H and determine which graph(s) G on n vertices and m edges maximize hom(G,H). We prove that if H is loop-threshold, then, for every n and m, there is a threshold graph G with n vertices a...

Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. The main motivating problem is the following: in the family of d-regular graphs, which graph G maximize/minimize the quantity i(G), the number of independent sets in G normalized exponentially by the size of G? What if i(G) is replaced by some other grap...

Eccentric Connectivity Index: Extremal Graphs and Values

Eccentric connectivity index has been found to have a low degeneracy and hence a significant potential of predicting biological activity of certain classes of chemical compounds. We present here explicit formulas for eccentric connectivity index of various families of graphs. We also show that the eccentric connectivity index grows at most polynomially with the number of vertices and determine ...

Extremal graphs for weights